"... students don't have misconceptions."

Here is an illustration of a very provocative argument.

Rochelle Gutierrez goes big in her introduction to Rehumanizing Mathematics for Black, Indigenous, and Latinx Students:

… students don’t have misconceptions. They have conceptions. And those conceptions make sense for them, until they encounter something that no longer works. They are only “misconceptions” when we begin with the expectation that others need to come to our way of thinking or viewing the world.

I saw an illustration of Gutierrez’s quote in a lesson where a bunch of students had been successfully graphing inequalities all morning until they crashed on a new one:

Over the last ten years, math education has done a top-shelf job destigmatizing making mistakes and revealing misconceptions. But when teachers fixate on the idea of a student having a misconception, even if to say that’s okay!, they are fixated on what they themselves know. 

Psychologist David Ausubel said, “The most important single factor influencing learning is what a learner already knows.” So Gutierrez’s critique isn’t purely social here; it’s also cognitive and pedagogical. Our teaching is less effective when we’re focused on what we know and more effective when we focus on what students know.

What do these students likely know?

• There is a connection between the inequality and the graph.

• They both describe a bunch of solutions.

• There is a critical point on the graph where solutions turn into not-solutions.

What I did next:

Gutierrez:

And those conceptions make sense for them, until they encounter something that no longer works.

These students don’t have a misconception. They have a conception, an idea, that whatever number is by itself on one side of an inequality, that’s the critical point. They need to encounter something that helps them find the limits of that idea. It works sometimes but they’ve overgeneralized it.

So as a class we evaluated x = 1 and convinced ourselves that it definitely works in the inequality and the graph. I used my authority to herald the fact that this is a very good thing, yea verily, these should match!

Then we evaluated x = 20 and convinced ourselves that it definitely works in the graph but definitely not in the inequality where it produces the false statement 34 ≤ 25. Then I asked students to discuss what they saw at their table, try again, and people found a new path.

PD is PD

Something exciting and a little unnerving about Gutierrez’s quote up there is that it applies equally to our classroom and our homes, our neighborhoods, our communities, our entire social lives. Everyone whose behavior befuddles us in some way—spouses, partners, friends, neighbors, etc—offers us the opportunity to either focus on what we know and how we would have done it or to wonder why what someone else did made sense to them and what we’ll need to do to develop a different shared understanding. 

What I’m saying is that PD is also PD–professional development in teaching is personal development in life. This is not true for many, many jobs on offer. But because our work is social and very intensely personal, personal transformation results in professional transformation and vice versa. I’d like this work to transform me into a more loving, curious individual and my life to do the same for my teaching, and starting with the conviction that “students don’t have misconceptions,” is one such transformation.